Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$.

How does one prove that weak solutions are viscosity solutions and vice versa for the problem $$ \begin{cases} -\Delta u = f(x) & \text{ in } \Omega\\ u=g & \text{ on } \partial \Omega \end{cases} $$ under suitable assumptions (to be determined) on $f$ and $g$?

The general proof is in this paper, but I'm looking for a simple proof in the case of this toy problem. I wouldn't want to rely on deep general results on elliptic regularity.

strictsub/super solution is trivial from the definition of viscosity solutions. $\endgroup$